Just want to make sure I am looking at this right.

You use a move action to add movement points to your pool based on your speed.

This is what I want to make sure of:

Something adds to or subtracts from your speed after adding the movement points to your pool, that doesn't affect the points in your pool.

Example:

Luke uses a move action to add 5 movement points to his pool.

Next, he picks up a crate that gives him -2 to his speed.

Then he moves 5 with the crate.

Picking up the crate affects his speed stat, not the movement points in the pool.

Correct.

In your example, subsequent move actions while holding the crate would give Luke 3 movement points per move action instead of 5.

Note - Usually only heroes can interact with objects like crates, unless otherwise specifically stated by the scenario setup.

This is a skirmish related question there is no heroes. And you don't need to interact to pick them up.

I believe the top question/interpretation is correct.

However I think Toqtamish's don't need to interact is wrong. I think you do have to interact to pick them up. Mission tokens are considered objects and in order to retrieve a token you must interact with it. To interact with it, you must use an action. Digging through the RRG can be a chore sometimes, hope I got this right.

Page20 of the RRG

Objects
Objects refer to elements on the map that are not figures. Doors,
crates, and terminals are all considered to be objects.
• Most mission tokens are considered to be objects, unless the
mission rules use the token to represent a figure.

Page 22 of the RRG

Retrieving Tokens
Some missions allow figures to retrieve specific tokens. A figure
adjacent to or in the same space as the token can perform an
interact to retrieve the token. Place the token on the figure’s base
to mark that the figure is now carrying that token. The token now
moves with the figure.

Page 15 of the RRG

Interact
Figures can interact to perform a variety of different tasks. For
example, a figure can open a door by performing an interact.
• A figure can perform an interact as an action.
• Mission rules will often list different tasks that can be
performed by interacting, such as using terminals or retrieving
tokens.

You have to perform an Interact action to retrieve/pick-up a token, but there's no action to drop it off. So 1 mission says to be adjacent to the terminal, and another says to be in your deployment zone at the end of the round. Neither would require an action to drop the token for points.